Problem: The three points $(3,-5)$, $(-a + 2, 3)$, and $(2a+3,2)$ lie on the same line.  What is $a$?
Answer: Because the three points are on the same line, the slope between the first and second equals the slope between the first and the third.  This gives us the equation: \begin{align*}
\frac{3-(-5)}{(-a+2) -3} &= \frac{2- (-5)}{(2a+3) - 3} \\
\frac{8}{-a-1} &= \frac{7}{2a} \\
8(2a) &= 7(-a-1) \\
23a &= -7 \\
&a = \boxed{\frac{-7}{23}}.
\end{align*}